“As being is to become, so is truth to belief” (Plato, “Timaeus”, cca. 360 BC)
“The first duty of man is the seeking after and the investigation of truth.” (Marcus Tullius Cicero, “De Officiis”, [“On Duties”], cca. 44 BC)
“The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another.” (Marcus Tullius Cicero, “De officiis” [“On Duties”], cca.44 BC)
“Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.” (Marcus Aurelius, "Meditations", cca. 2nd century)
“We are meant to take them [the words ‘increase and multiply’] in a figurative sense. […] It is only in the case of signs outwardly given that we find increase and multiplication in the sense that a single truth can be expressed by several different means […] that a single expression can be interpreted in several different ways.” (St. Augustine, “Confessions”, 397- 400)
"Truth is sought for itself, but the truths are immersed in uncertainties." (Abu Ali al-Hasan ibn al-Haytham [Alhazen], "Aporias against Ptolemy", 1025-1028)
“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” (Roger Bacon, “Opus Majus” Book 1, 1267)
"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience."(Roger Bacon, "Opus Majus", cca. 1267)
“The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you.” (Richard de Burry, “Philobiblon”, 1344)
"Man's mind is so formed that it is far more susceptible to falsehood than to truth." (Desiderius Erasmus, "Praise of Folly", 1509)
“[…] no pleasure is comparable to the standing upon the vantage ground of truth […]” (Sir Francis Bacon, “Essays”, 1597)
“The first and most ancient inquirers into truth were wont to throw their knowledge into aphorisms, or short, scattered, unmethodical sentences.” (Lord Bacon, “Novum Organum”, 1620)
"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)
“[…] thus each truth discovered was a rule available in the discovery of subsequent ones.” (René Descartes, “Discourse on Method”, 1637)
“Every man is not a proper champion for truth, nor fit to take up the gauntlet in the cause of verity: many from the ignorance of these maxims, and an inconsiderate zeal for truth, have too rashly charged the troops of error, and remain as trophies unto the enemies of truth. A man may be in as just possession of truth as of a city, and yet be forced to surrender: ’tis therefore far better to enjoy her with peace than to hazard her on a battle: if therefore there rise any doubts in my way, I do forget them, or at least defer them, till my better settled judgment and more manly reason be able to resolve them.” (Sir Thomas Browne, ”Religio Medici”, 1643)
“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.” (René Descartes, “Principles of Philosophy”, 1644)
“Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know.” (Sir Thomas Browne, “Pseudodoxia Epidemica”, 1646)
“Men are apt to prefer a prosperous error before an afflicted truth.” (Jeremy Taylor, “The Rule and Exercises of Holy Living”, 1650)
“All things being double-handed, and having the appearances both of truth and falsehood, where our affections have engaged us we attend only to the former.” (Joseph Glanvill, “Scepsis”, 1665)
“Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” (Blaise Pascal, “Pensées”, 1670)
“We see neither justice nor injustice which does not change its nature with change in climate. Three degrees of latitude reverse all jurisprudence: a meridian decides the truth.” (Blaise Pascal, “Pensées”, 1670)
“In logic, they teach that contraries laid together more evidently appear: it follows, then, that all controversy being permitted, falsehood will appear more false, and truth the more true; which must needs conduce much to the general confirmation of an implicit truth.” (John Milton, “True Religion, Heresy, Schism, Toleration, and what best means may be used against the Growth of Popery”, 1673)
“In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow.” (Baruch Spinoza, [letter to Hugo Boxel], 1674)
“Truth is always consistent with itself, and needs nothing to help it out; it is always near at hand, and sits upon our lips, and is ready to drop out before we are aware; whereas a lie is troublesome, and sets a man’s invention upon the rack, and one trick needs a great many more to make it good.” (John Tillotson, “Sermons”, 1682)
"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)
"Two things are identical if one can be substituted for the other without affecting the truth." (Gottfried W Leibniz, "Table de definitions", 1704)
“Not only the investigation of truth, but the communication of it also, is often practised in such a method as neither agrees precisely to synthetic or analytic.” (Isaac Watts, “Logic, or The right use of reason, in the inquiry after truth”, 1725)
“He that would make a real progress in knowledge must dedicate his age as well as first fruits - the latter growth as well as the first-fruits - at the altar of truth.” (Bishop George Berkeley, “Siris”, 1744)
“If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty.” (Edmund Burke, “Essay on the Sublime and Beautiful”, 1756)
"There are truths which are not for all men, nor for all times." (Voltaire, [Letter to François-Joachim de Pierre] 1764)
"Ignorance is preferable to error; and he is less remote from the truth who believes nothing, than he who believes what is wrong." (Thomas Jefferson, "Notes on the State of Virginia", 1781)
"General abstract truth is the most precious of all blessings; without it, man is blind; it is the eye of reason." (Jean-Jacques Rousseau, "The Confessions of J. J. Rousseau", 1783)
“The discovery of truth by slow, progressive meditation is talent. Intuition of the truth, not preceded by perceptible meditation, is genius.” (Johann K Lavater, 1787)
“Everything possible to be believed is an image of truth.” (William Blake, “The Marriage of Heaven and Hell”, 1790)
“If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world.” (Immanuel Kant, “Critique of Judgment”, 1790)
"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoine Lavoisier, "Elements of Chemistry", 1790)
"It is an acknowledged truth in philosophy that a just theory will always be confirmed by experiment." (Thomas R Malthus, "An Essay on The Principle of Population", 1798)
“Forgetting that the only eternal part for man to act is man, and that the only immutable greatness is truth.” (Alphonse Lamartine, “The History of the Restoration of Monarchy in France”, 1851)
"Accuracy of language is one of the bulwarks of truth." (Anna B Jameson, "A Commonplace Book of Thoughts, Memories, and Fancies", 1854)
"We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis." (James C Maxwell, "On Faraday’s lines of force", 1855)
"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)
“The excellence of every art is its intensity, capable of making all disagreeables evaporate from their being in close relationship with beauty and truth.” (John Keats. [letter to George and Thomas Keats] 1817)
"[...] all knowledge, and especially the weightiest knowledge of the truth, to which only a brief triumph is allotted between the two long periods in which it is condemned as paradoxical or disparaged as trivial." (Arthur Schopenhauer, "The World as Will and Representation", 1819)
"We are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it." (Thomas Jefferson, [Letter to William Roscoe] 1820)
"Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric. For both of these, nothing has value but form; content is immaterial. Whether mathematics is adding up pennies or guineas, whether rhetoric is defending truth or falsehood, makes no difference to either.” (Johann Wolfgang von Goethe, "Wilhelm Meisters Wanderjahre" ["Reflections in the Spirit of the Wanderers"], 1821)
“Facts are the mere dross of history. It is from the abstract truth which interpenetrates them, and lies latent among them, like gold in the ore, that the mass derives its whole value: and the precious particles are generally combined with the baser in such a manner that the separation is a task of the utmost difficulty.” (Thomas B Macaulay, “History”, 1828)
"Truth in itself is rarely sufficient to make men act. Hence the step is always long from cognition to volition, from knowledge to ability. The most powerful springs of action in men lie in his emotions." (Carl von Clausewitz, "On War", 1832)
“It is difficult to discriminate the voice of truth from amid the clamour raised by heated partisans.” (Friedrich Schiller, “Schillers Sammtliche Werke”, 1834)
“The most important and lasting truths are the most obvious ones. Nature cheats us with her mysteries, one after another, like a juggler with his tricks; but shews us her plain honest face, without our paying for it.” (William Hazlitt, “Characteristics: In the Manner of Rochefoucault's Maxims”, 1837)
“In truth, ideas and principles are independent of men; the application of them and their illustration is man's duty and merit.” (Edward Forbes, 1847)
“The peculiarity of the evidence of mathematical truths is that all the argument is on one side.” (John Stuart Mill, “On Liberty”, 1859)
[…] the besetting danger is not so much of embracing falsehood for truth, as of mistaking a part of the truth for the whole.” (John Stuart Mill, “Dissertations and Discussions: Political, Philosophical, and Historical”, 1864)
"No departure from the truth of nature shall be discovered by the closest scrutiny." (Henry P Robinson, "Pictorial Effect in Photography", 1869)
"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)
“Pure truth cannot be assimilated by the crowd; it must be communicated by contagion.” (Henri-Frédéric Amiel, [journal entry] 1875)
"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)
“Convictions are more dangerous enemies of truth than lies.” (Friedrich Nietzsche, “Human, All Too Human: A book for Free Spirits”, 1878)
“It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors.” (Thomas H Huxley, “The Progress of Science”, 1887)
“How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?” (Sir Arthur Conan Doyle, “The Sign of Four”, 1890)
“Accuracy of statement is one of the first elements of truth; inaccuracy is a near kin to falsehood.” (Tyron Edwards, “A Dictionary of Thoughts”, 1891)
"There is no short cut to truth, no way to gain a knowledge of the universe except through the gateway of scientific method." (Karl Pearson, “The Grammar of Science”, 1892)
"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)
"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valery, "Introduction to the Method of Leonardo da Vinci", 1895)
"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notations, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. (William S Jevons, "The Principles of Science" 2nd Ed., 1900)
"The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1900)
“Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.” (Henri Poincaré, “Science and Hypothesis”, 1902)
“Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic - for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it.” (Gilbert K Chesterton, Daily News, 1905)
"The forceps of our minds are clumsy forceps, and crush the truth a little in taking hold of it." (Herbert G Wells, "Scepticism of the Instrument: A Modern Utopia", 1905)
“The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason.” (William E Chancellor, “A Theory of Motives, Ideals and Values in Education” 1907)
“The truth of an idea is not a stagnant property inherent in it. Truth happens to an idea. It becomes true, is made true by events. Its verity is in fact an event, a process: the process namely of its verifying itself, its verification. Its validity is the process of its validation.” (William James, “Pragmatism: A New Name for Some Old Ways of Thinking”, 1907)
"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)
"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest groimds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, "A Plea for the Mathematician", Nature, 1908)
“Truth is on a curve whose asymptote our spirit follows eternally.” (Léo Errera, “Recueil d'Œuvres de Léo Errera: Botanique Générale”, 1908)
“Science is not the monopoly of the naturalist or the scholar, nor is it anything mysterious or esoteric. Science is the search for truth, and truth is the adequacy of a description of facts.” (Paul Carus, “Philosophy as a Science”, 1909)
“The pursuit of truth is chimerical. […] There is no permanent absolute unchangeable truth; what we should pursue is the most convenient arrangement of our ideas.” (Samuel Butler, “Notebooks”, 1912)
“Only in men’s imagination does every truth find an effective and undeniable existence.” (Joseph Conrad, “Some Reminiscences”, 1912)
“The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics.” (J Craig, “A Course of Study for the Preparation of Rural School Teachers”, 1912)
"The conception of logical laws must be the decisive factor in the treatment of logic, and that conception depends upon what we understand by the word ‘true’. It is generally admitted at the very beginning that logical laws must be rules of conduct to guide thought to truth […]" (Gottlob Frege," Grundgesetze", The Monist, 1915)
“As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies. “ (Albert Einstein: “Relativity: The Special and General Theory”, 1916)
“It may be impossible for human intelligence to comprehend absolute truth, but it is possible to observe Nature with an unbiased mind and to bear truthful testimony of things seen.” (Sir Richard A Gregory, “Discovery, Or, The Spirit and Service of Science”, 1916)
“[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination.” (James B Shaw, “Lectures on the Philosophy of Mathematics”, 1918)
“Ignorance may find a truth on its doorstep that erudition vainly seeks in the stars.” (George Iles, “Canadian Stories”, 1918)
"It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability." (John M Keynes, "A Treatise on Probability", 1921)
"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)
"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)
“We all know that Art is not truth. Art is a lie that makes us realize truth.” (Pablo Picasso, “The Arts”, 1923)
“Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved.” (Bertrand Russell, “The ABC of Relativity”, 1925)
"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making", 1926)
"The scientist is a lover of truth for the very love of truth itself, wherever it may lead." (Luther Burbank, "Why I Am An Infidel", 1926)
“If our so-called facts are changing shadows, they are shadows cast by the light of constant truth.” (Sir Arthur S Eddington, “Science and the Unseen World”, 1929)
“Try to be conspicuously accurate in everything, pictures as well as text. Truth is not only stranger than fiction, it is more interesting.” (William R Hearst, “Letter of Instruction to Hearst Publishers”, 1929)
“Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man.” (Bertrand Russell, “The Scientific Outlook”, 1931)
“It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him.” (Max Planck, “Where is Science Going?”, 1932)
“Apart from blunt truth, our lives sink decadently amid the perfume of hints and suggestions.” (Alfred N Whitehead, “Adventures of Ideas”, 1933)
”[…] the merit of mathematics, in all its forms, consists in its truth; truth conveyed to the understanding, not directly by words but by symbols which serve as the world’s only universal written language.” (David Eugene Smith, “The Poetry of Mathematics and Other Essays”, 1934)
“Mathematics is the science of number and space. It starts from a group of self-evident truths and by infallible deduction arrives at incontestable conclusions […] the facts of mathematics are absolute, unalterable, and eternal truths.” (E Russell Stabler, “An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics”, The Mathematics Teacher Vol 26, 1935)
"Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1938)
"Even if all parts of a problem seem to fit together like the pieces of a jigsaw puzzle, one has to remember that the probable need not necessarily be the truth and the truth not always probable." (Sigmund Freud, "Moses and Monotheism", 1939)
“When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic.” (Jean Rostand, “Pensées d'un Biologiste”, 1939)
“A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling.” (Robert Musil, “Man Without Qualities”, 1943)
"Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans," Physics and Philosophy" 3rd Ed., 1943)
"Thus we do not try to prove the existence of the external world – we discover it, because the fundamental power of words or other symbols to represent events [...] permits us to put forward hypotheses and test their truth by reference to experience." (Kenneth Craik, "The Nature of Explanation", 1943)
“When two hypotheses are possible, we provisionally choose that which our minds adjudge to the simpler on the supposition that this Is the more likely to lead in the direction of the truth.” (James H Jeans, “Physics and Philosophy” 3rd Ed., 1943)
"After all, the ultimate goal of all research is not objectivity, but truth." (Helene Deutsch, "The Psychology of Women", 1944)
"The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists." (Erwin Schrödinger, „What is Life?", 1944)
“I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.” (John von Neumann, "The Mathematician", The Works of the Mind Vol. I (1), 1947)
"Science condemns itself to failure when, yielding to the infatuation of the serious, it aspires to attain being, to contain it, and to possess it; but it finds its truth if it considers itself as a free engagement of thought in the given, aiming, at each discovery, not at fusion with the thing, but at the possibility of new discoveries; what the mind then projects is the concrete accomplishment of its freedom." (Simone de Beauvoir, "The Ethics of Ambiguity", 1947)
"Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree.” (Norbert Wiener, “Cybernetics”, 1948)
"A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." (Max Planck, "A Scientific Autobiography", 1949)
“Representation of the world, like the world itself, is the work of men; they describe it from their own point of view, which they confuse with absolute truth.” (Simone de Beauvoir, “The Second Sex”, 1949)
"Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)
"It is true that the grasping of truth is not possible without empirical basis. However, the deeper we penetrate and the more extensive and embracing our theories become the less empirical knowledge is needed to determine those theories." (Albert Einstein, 1952)
“Logic and truth are two very different things, but they often look the same to the mind that’s performing the logic. " (Theodore Sturgeon, “More Than Human”, 1953)
"We cannot define truth in science until we move from fact to law. And within the body of laws in turn, what impresses us as truth is the orderly coherence of the pieces. They fit together like the characters of a great novel, or like the words of a poem. Indeed, we should keep that last analogy by us always, for science is a language, and like a language it defines its parts by the way they make up a meaning. Every word in a sentence has some uncertainty of definition, and yet the sentence defines its own meaning and that of its words conclusively. It is the internal unity and coherence of science which gives it truth, and which makes it a better system of prediction than any less orderly language." (Jacob Bronowski, "The Common Sense of Science", 1953)
"There are no whole truths; all truths are half-truths. It is trying to treat them as whole truths that plays the devil.” (Alfred North Whitehead, “Dialogues”, 1954)
"The history of science is rich in the example of the fruitfulness of bringing two sets of techniques, two sets of ideas, developed in separate contexts for the pursuit of new truth, into touch with one another." (J. Robert Oppenheimer, "Science and the common understanding", 1954)
"Science is the creation of concepts and their exploration in the facts. It has no other test of the concept than its empirical truth to fact." (Jacob Bronowski, "Science and Human Values", 1956)
“Starting from statistical observations and applying to them a clear and precise concept of probability it is possible to arrive at conclusions which are just as reliable and ‘truth-full’ and quite as practically useful as those obtained in any other exact science.” (Richard von Mises, “Probability, Statistics, and Truth”2nd Ed., 1957)
“Uncertainty is introduced, however, by the impossibility of making generalizations, most of the time, which happens to all members of a class. Even scientific truth is a matter of probability and the degree of probability stops somewhere short of certainty.” (Wayne C Minnick, “The Art of Persuasion”, 1957)
"We speak in terms of ‘acceptance’, ‘confidence’, and ‘probability’, not ‘proof’. If by proof it is meant the establishment of eternal and absolute truth, open to no possible exception or modification, then proof has no place in the natural sciences." (George G Simpson, “Life: An Introduction to Biology”, 1957)
“We can never achieve absolute truth but we can live hopefully by a system of calculated probabilities. The law of probability gives to natural and human sciences - to human experience as a whole - the unity of life we seek.” (Agnes E Meyer, “Education for a New Morality”, 1957)
"It will never be possible by pure reason to arrive at some absolute truth." (Werner K Heisenberg, "Physics and Philosophy: The revolution in modern science", 1958)
"Scientific method is the way to truth, but it affords, even in principle, no unique definition of truth. Any so-called pragmatic definition of truth is doomed to failure equally." (Willard v O Quine, "Word and Object", 1960)
"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)
"Relativity is inherently convergent, though convergent toward a plurality of centers of abstract truths. Degrees of accuracy are only degrees of refinement and magnitude in no way affects the fundamental reliability, which refers, as directional or angular sense, toward centralized truths. Truth is a relationship." (R Buckminster Fuller, "The Designers and the Politicians", 1962)
"When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic." (Jean Rostand, "The substance of man", 1962)
“Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected.” (Richard Feynman, “The Feynman Lectures on Physics” Vol. 1,1964)
“[…] in the statistical world you can multiply ignorance by a constant and get truth.” (Raymond F Jones, “The Non-Statistical Man”, 1964)
"The belief that there is only one truth and that oneself is in possession of it, seems to me the deepest root of all that is evil in the world." (Max Born, "Natural Philosophy of Cause and Chance", 1964)
“The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links.” (Arthur Koestler, “The Act of Creation”, 1964)
“All views are only probable, and a doctrine of probability which is not bound to a truth dissolves into thin air. In order to describe the probable, you must have a firm hold on the true. Therefore, before there can be any truth whatsoever, there must be absolute truth.” (Jean-Paul Sartre, “The Philosophy of Existentialism”, 1965)
“Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.” (Salomon Bochner, “Why Mathematics Grows”, Journal of the History of Ideas, 1965)
“[…] truth is the intersection of independent lies.” (Richard Levins, “The Strategy of Model Building in Population Biology”, 1966)
"Primary scientific papers are not meant to be final statement of indisputable truths; each is merely a tiny tentative step forward, through the jungle of ignorance." (Erwin Schrödinger, "Information, Communication, Knowledge", Nature Vol. 224 (5217), 1969)
"At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too." (Willard v O Quine, "The Web of Belief", 1970)
"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn," The Structure of Scientific Revolutions", 1970)
“Probability is truth in some degree […]” (Errol E Harris, “Hypothesis and Perception: The Roots of Scientific Method”, 1970)
“In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.” (Martin Gardner, “Mathematical Games”, Scientific American, 1973)
“No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect.” (Arthur C Clarke, “Profiles of the Future”, 1973)
“No matter how much reverence is paid to anything purporting to be ‘statistics’, the term has no meaning unless the source, relevance, and truth are all checked.” (Tom Burnam, “The Dictionary of Misinformation”, 1975)
“It seems that truth
Is progressive approximation
In which the relative fraction
Of our spontaneously tolerated residual error
Constantly diminishes.” (R Buckminster Fuller, “And It Came to Pass - Not to Stay”, 1976)
„[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)
"Truth cannot be defined or tested by agreement with 'the world'; for not only do truths differ for different worlds but the nature of „[…] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility.“ (Lynn A Steen, „Mathematics Today: Twelve Informal Essays“, 1978)
“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous.” (Nelson Goodman, “Ways of Worldmaking”, 1978)
"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.” (Stephen J Gould, “The Mismeasure of Man”, 1980)
"[…] the truth or likeness to truth that much of science pursues is of a rather special kind – we might call it 'physically necessary truth'" (L Jonathan Cohen, "What has science to do with truth?", Synthese 45, 1980)
"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, “The Role of Mathematics in the Rise of Science”, 1981)
"True, the initial ideas are in general those of an individual, but the establishment of the reality and truth is in general the work of more than one person." (Willard Libby, "Talking to people", 1981)
“In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it.” (Rudy Rucker, “Infinity and the Mind: The science and philosophy of the infinite”, 1982)
"Scientific theories must tell us both what is true in nature, and how we are to explain it. […] Scientific theories are thought to explain by dint of the descriptions they give of reality." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)
"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, Vol. 5, No. 4, 1983)
"The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension." (Paul R Halmos, “I Want to Be a Mathematician”, 1985)
"There is no coherent knowledge, i.e. no uniform comprehensive account of the world and the events in it. There is no comprehensive truth that goes beyond an enumeration of details, but there are many pieces of information, obtained in different ways from different sources and collected for the benefit of the curious. The best way of presenting such knowledge is the list - and the oldest scientific works were indeed lists of facts, parts, coincidences, problems in several specialized domains." (Paul K Feyerabend, “Farewell to Reason”, 1987)
“Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it.” (Isaac Asimov, [Interview by Bill Moyers] 1988)
“[…] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.” (Ivars Peterson, “Islands of Truth”, 1990)
"It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty.” (Mahlon B Hoagland, “Toward the Habit of Truth”, 1990)
"It is not merely the truth of science that makes it beautiful, but its simplicity.” (Walker Percy, “Signposts in a Strange Land”, 1991)
„[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less ‚beautiful‘. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula.“ (Jean Dieudonné, „Mathematics - The Music of Reason“, 1992)
“Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth.” (Clifford Truesdell and Walter Noll, “The Non-Linear Field Theories of Mechanics” 2nd Ed., 1992)
"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero, “The Man Who Counted”, 1993)
“The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’.” (Richard Feynman, “Six Easy Pieces”, 1994)
“[…] equations are like poetry: They speak truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend.” (Michael Guillen, “Five Equations That Changed the World”, 1995)
“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)
“Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith.” (K C Cole, “The Universe and the Teacup: The Mathematics of Truth and Beauty”, 1997)
"Mathematical logic deals not with the truth but only with the game of truth.” (Gian-Carlo Rota, “Indiscrete Thoughts”, 1997)
“Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)
“Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net.” (John Polkinghorne, “Belief in God in an Age of Science”, 1998)
“Mathematics has no privileged road to the truth.”(Donald C Benson, “The Moment of Proof: Mathematical Epiphanies”, 1999)
“Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!” (Gregory J Chaitin, “A century of controversy over the foundations of mathematics”, 2000)
"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)
“Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.” (Yōko Ogawa, "The Housekeeper and the Professor", 2003)
“A model is a simplification or approximation of reality and hence will not reflect all of reality. […] Box noted that ‘all models are wrong, but some are useful’. While a model can never be ‘truth’, a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless.” (Kenneth P Burnham & David R Anderson, “Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach” 2nd Ed., 2005)
"It is art which invents the lies that raise falsehood to its highest affirmative power, that turns the will to deceive into something which is affirmed in the power of falsehood. For the artist, appearance no longer means the negation of the real in this world but this kind of selection, correction, redoubling and affirmation. Then truth perhaps takes on a new sense. Truth is appearance." (Gilles Deleuze, "Nietzsche as Philosopher", 2005)
“Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception.“ (David Ruelle, “The Mathematician's Brain”, 2007)
"It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject.” (Steven G Krantz, "The History and Concept of Mathematical", 2007)
“Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty.” (William Byers, “How Mathematicians Think”, 2007)
“Geometrical truth is (as we now speak) synthetic: it states facts about the world. Such truths are not ordinary truths but essential truths, giving the reality of the empirical world in which they are imperfect embodied.” (Fred Wilson, “The External World and Our Knowledge of It”, 2008)
"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’.” (Vladimir G Ivancevic & Tijana T Ivancevic, “Quantum Leap”, 2008)
"Mathematicians, like priests, seek ‘ideal’, immutable truths and then often try to apply these truths to the real world." (Clifford A Pickover, "The Loom of God: Tapestries of Mathematics and Mysticism", 2009)
"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)
"There is an absolute nature to truth in mathematics, which is unmatched in any other branch of knowledge. A theorem, once proven, requires independent checking but not repetition or independent derivation to be accepted as correct. […] Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity” (James Glimm, "Reflections and Prospectives", 2009)
“A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms.” (Raymond S Nickerson, “Mathematical Reasoning”, 2010)
“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs”, 2010)
“A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked.” (Sara Negri & Jan von Plato, “Proof Analysis”, 2011)
“[…] statistics is a method of pursuing truth. At a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This type of pursuit of truth, especially in the form of an event’s future likelihood, is the essence of psychology, of science, and of human evolution.” (Arthur Aron et al, "Statistics for Psychology" 6th Ed., 2012)
“Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.” (Edward Frenkel, “Love and Math”, 2014)
"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)
“Scientists generally agree that no theory is 100 percent correct. Thus, the real test of knowledge is not truth, but utility.” (Yuval N Harari, “Sapiens: A brief history of humankind”, 2017)
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